
equation (REF ) holds in \(D^{\prime }(0,T;\Omega )\)  and
\(P(\varrho ), \varrho |v|^2\in L^\infty (0,T;L^1(\Omega )),\) 

the density \(\varrho \)  is a renormalized solution of (REF ) in the sense of [1]}.

the energy inequality holds
\(\begin{aligned}E(t) \le E(0), \end{aligned}\) 
where \(E(t)=\int _{\Omega }\left( \frac{1}{2}\varrho |v|^2+\kappa {\gamma -1} \right) dx\) .


